New York (CNN) -- With the clock ticking on New York's legislative session, Gov. Andrew Cuomo is hoping lawmakers will use their last scheduled day to take up his proposed legislation to legalize same-sex marriage.

Currently, 31 senators, including two Republicans, are in favor of the bill. Its backers need one more GOP member to vote in favor for it to pass.

Cuomo says it would grant same-sex couples equal rights to marry "as well as hundreds of rights, benefits and protections that are currently limited to married couples of the opposite sex."

The first-term governor has lobbied opposition and undecided senators in an effort to secure the single vote needed for its passage.

A vote on the measure, which the state Assembly passed Wednesday night, has been stalled in part by Republican concerns over protections for religious institutions against the potential for litigation in the wake of the proposed law.

Republicans, led by Senate Majority Leader Dean Skelos, have expressed concerns over the "unintended consequences" of a bill that redefines the legal parameters of marriage.

"There have been some meetings with the governor's office indicating that they are receptive to some changes to the legislation," Skelos said in Albany last week, indicating discussions would continue.

Democrat Tom Duane, an openly gay member of the state Senate, countered that there are "already laws on the books in New York state which protect people of faith."

"Marriage is a civil right. People get married, and it is recognized in government," he said. "We're talking about the government recognizing same-sex equality."

Still, some analysts contend that Cuomo's discussions with Republican lawmakers could be intended to provided political cover for those who want to vote for the measure but are dissuaded by the potential backlash that could result from breaking ranks with their conservation base.

"I don't think any one Republican wants to the one person deciding the vote ... because that person would not get a lot of conservative support in the next election," said Kenji Yoshino, a professor of constitutional law at New York University's School of Law.

"Ironically, it might be easier to get 35 (votes) than it is to get 32," he added.

Cuomo has the option of extending the legislative session and has indicated he's willing to do just that.

"There is a full agenda for both the Assembly and Senate to accomplish and the legislative session will not end, either through regular or special session, until the people's business is done," the governor said last week.

A recent Siena Research Institute poll found that 58% of New Yorkers support same-sex marriage, while more than one-quarter of voters say the measure -- along with extending rent regulation laws -- is one of their top two priorities.

Despite the polling numbers, a group opposed to same-sex marriage says New York lawmakers shouldn't be making the ultimate decision; the voters should.

"The voters of New York should have the same ability as 31 other states," Brian Brown, president of National Organization for Marriage, told CNN's Don Lemon Sunday, saying a referendum in favor of same-sex marriage has always failed.

"We know what marriage is," he said. "We know that it's unique and special, and we do not want it redefined."

But Daniel O'Donnell, a member of the New York State Assembly who favors same-sex marriage said the issue is about equality.

"The Supreme Court has ruled that marriage is one of our fundamental rights," he told Lemon. "I'm not seeking a marriage in the church ... I'm not seeking anything from any religion."

History hasn't been kind to attempts to pass same-sex marriage in New York.

The Senate rejected a bill to legalize same-sex marriage in 2009 and has blocked the last three attempts by the state's lower house to get the proposed legislation signed into law.

Although New York currently does not grant same-sex marriages, a 2008 appellate court ruling upheld the right of couples to have their same-sex marriages recognized if they are performed elsewhere.

The current measure would amend the state's Domestic Relations Law to say, among other changes, that "no application for a marriage license shall be denied on the ground that the parties are of the same or a different sex."

Five states -- Massachusetts, Connecticut, Iowa, Vermont and New Hampshire -- and the District of Columbia currently grant same-sex marriage licenses.

CNN's Ed Payne, David Ariosto and Dana Garrett contributed to this report.